Question

Question 10 (b): Show by example that part (a) may be false if $f$ is not surjective.

Short answer

Answer:

It can be proved by a counter example.

Step-by-step solution

Supposition of a homomorphic map

Consider the inclusion map$\theta :\mathrm{ℝ}\to \mathrm{ℂ}$

It is a homomorphism of fields but is not surjective.

Applying the condition

The field is an $\mathrm{ℝ}$ideal itself.

$\theta \left(\mathrm{ℝ}\right)=\mathrm{ℝ}$

But$\mathrm{ℝ}$ is not an ideal in $\mathrm{ℂ}$.

Conclusion

It is false because f is not surjective.